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Evolution from helical to collinear ferromagnetic order of the Eu$^{2+}$ spins in RbEu(Fe$_{1-x}$Ni$_{x}$)$_{4}$As$_{4}$

The ground-state magnetic structures of the Eu$^{2+}$ spins in recently discovered RbEu(Fe$_{1-x}$Ni$_{x}$)$_{4}$As$_{4}$ superconductors have been investigated by neutron powder diffraction measurements. It is found that as the superconductivity gets suppressed with the increase of Ni doping, the magnetic propagation vector of the Eu sublattice diminishes, corresponding to the decrease of the rotation angle between the moments in neighboring Eu layers. The ferromagnetic Eu layers are helically modulated along the $\mathit{c}$ axis with an incommensurate magnetic propagation vector in both the ferromagnetic superconductor RbEu(Fe$_{0.95}$Ni$_{0.05}$)$_{4}$As$_{4}$ and the superconducting ferromagnet RbEu(Fe$_{0.93}$Ni$_{0.07}$)$_{4}$As$_{4}$. Such a helical structure transforms into a purely collinear ferromagnetic structure for non-superconducting RbEu(Fe$_{0.91}$Ni$_{0.09}$)$_{4}$As$_{4}$, with all the Eu$^{2+}$ spins lying along the tetragonal (1 1 0) direction. The evolution from helical to collinear ferromagnetic order of the Eu$^{2+}$ spins with increasing Ni doping is supported by first-principles calculations. The variation of the rotation angle between adjacent Eu$^{2+}$ layers can be well explained by considering the change of magnetic exchange couplings mediated by the indirect Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction.